The second part of the claim is straightforward from (REF ); we thus have to prove that \(\phi \)  is a projective trivial fibration.
To that end, we check that \(\phi \)  has the right lifting property with respect to the morphisms \({{Y}}_c \otimes \partial \Delta ^n \rightarrow {{Y}}_c \otimes \Delta ^n\)  for \(c \in {{C}\mathrm {art}}\)  and \(n \in \mathbb {N}_0\) .
These morphisms form a set of generating cofibrations for the projective model structure \({H}_\infty ^p\)  (see e.g. [1]}).
For any morphism \(p \colon Y \rightarrow X\)  of simplicial presheaves, there is a bijection between solutions to the lifting problems
\(\begin{tikzcd}{{Y}}_c \otimes \partial \Delta ^n [r] [d] & Y [d, "p"]\\{{Y}}_c \otimes \Delta ^n [r] & X\end{tikzcd}\qquad \text{and} \qquad \begin{tikzcd}\partial \Delta ^n [r] [d] & Y(c) [d, "p_{|c}"]\\\Delta ^n [r] & X(c)\end{tikzcd}\) 
